This sounds like a normal reaction on your cat's part, but aggravated by the fact that you separate them and give the cat less attention and the dog more.
First, make sure the cat has at least a couple of places to go that are out of the dog's reach. Shelves are good for this.
Now, the cat will hiss and maybe swat at anybody it's not used to. But if it can get on a high place and watch the dog, it will become interested.
The cat does not sound especially aggressive; so I wouldn't worry too much.
Chihuahuas are yappy and jumpy, so that is normal. Being so small, they learn to bluff. I recommend ordinary obedience procedures - teaching the dog "sit" and "down [lie down]" and how to walk on a slack leash. If you don't know how to train this dog, take an obedience class. So many people are afraid to discipline such a tiny dog! The result is often that they don't even teach the chihuahua the word "NO," and they may bite. Sure it's not a timber wolf, but this is not good. Chihuahuas can learn to behave the same as any dog.